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Theorem List for Metamath Proof Explorer - 40801-40900   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremetransclem32 40801* This is the proof for the last equation in the proof of the derivative calculated in [Juillerat] p. 12, just after equation *(6) . (Contributed by Glauco Siliprandi, 5-Apr-2020.)
(𝜑𝑆 ∈ {ℝ, ℂ})    &   (𝜑𝑋 ∈ ((TopOpen‘ℂfld) ↾t 𝑆))    &   (𝜑𝑃 ∈ ℕ)    &   (𝜑𝑀 ∈ ℕ0)    &   𝐹 = (𝑥𝑋 ↦ ((𝑥↑(𝑃 − 1)) · ∏𝑗 ∈ (1...𝑀)((𝑥𝑗)↑𝑃)))    &   (𝜑𝑁 ∈ ℕ0)    &   (𝜑 → ((𝑀 · 𝑃) + (𝑃 − 1)) < 𝑁)    &   𝐻 = (𝑗 ∈ (0...𝑀) ↦ (𝑥𝑋 ↦ ((𝑥𝑗)↑if(𝑗 = 0, (𝑃 − 1), 𝑃))))       (𝜑 → ((𝑆 D𝑛 𝐹)‘𝑁) = (𝑥𝑋 ↦ 0))

Theoremetransclem33 40802* 𝐹 is smooth. (Contributed by Glauco Siliprandi, 5-Apr-2020.)
(𝜑𝑆 ∈ {ℝ, ℂ})    &   (𝜑𝑋 ∈ ((TopOpen‘ℂfld) ↾t 𝑆))    &   (𝜑𝑃 ∈ ℕ)    &   (𝜑𝑀 ∈ ℕ0)    &   𝐹 = (𝑥𝑋 ↦ ((𝑥↑(𝑃 − 1)) · ∏𝑗 ∈ (1...𝑀)((𝑥𝑗)↑𝑃)))    &   (𝜑𝑁 ∈ ℕ0)       (𝜑 → ((𝑆 D𝑛 𝐹)‘𝑁):𝑋⟶ℂ)

Theoremetransclem34 40803* The 𝑁-th derivative of 𝐹 is continuous. (Contributed by Glauco Siliprandi, 5-Apr-2020.)
(𝜑𝑆 ∈ {ℝ, ℂ})    &   (𝜑𝑋 ∈ ((TopOpen‘ℂfld) ↾t 𝑆))    &   (𝜑𝑃 ∈ ℕ)    &   (𝜑𝑀 ∈ ℕ0)    &   𝐹 = (𝑥𝑋 ↦ ((𝑥↑(𝑃 − 1)) · ∏𝑘 ∈ (1...𝑀)((𝑥𝑘)↑𝑃)))    &   (𝜑𝑁 ∈ ℕ0)    &   𝐻 = (𝑘 ∈ (0...𝑀) ↦ (𝑥𝑋 ↦ ((𝑥𝑘)↑if(𝑘 = 0, (𝑃 − 1), 𝑃))))    &   𝐶 = (𝑛 ∈ ℕ0 ↦ {𝑐 ∈ ((0...𝑛) ↑𝑚 (0...𝑀)) ∣ Σ𝑘 ∈ (0...𝑀)(𝑐𝑘) = 𝑛})       (𝜑 → ((𝑆 D𝑛 𝐹)‘𝑁) ∈ (𝑋cn→ℂ))

Theoremetransclem35 40804* 𝑃 does not divide the P-1 -th derivative of 𝐹 applied to 0. This is case 2 of the proof in [Juillerat] p. 13 . (Contributed by Glauco Siliprandi, 5-Apr-2020.)
(𝜑𝑃 ∈ ℕ)    &   (𝜑𝑀 ∈ ℕ0)    &   𝐹 = (𝑥 ∈ ℝ ↦ ((𝑥↑(𝑃 − 1)) · ∏𝑗 ∈ (1...𝑀)((𝑥𝑗)↑𝑃)))    &   𝐶 = (𝑛 ∈ ℕ0 ↦ {𝑐 ∈ ((0...𝑛) ↑𝑚 (0...𝑀)) ∣ Σ𝑗 ∈ (0...𝑀)(𝑐𝑗) = 𝑛})    &   𝐷 = (𝑗 ∈ (0...𝑀) ↦ if(𝑗 = 0, (𝑃 − 1), 0))       (𝜑 → (((ℝ D𝑛 𝐹)‘(𝑃 − 1))‘0) = ((!‘(𝑃 − 1)) · (∏𝑗 ∈ (1...𝑀)-𝑗𝑃)))

Theoremetransclem36 40805* The 𝑁-th derivative of 𝐹 applied to 𝐽 is an integer. (Contributed by Glauco Siliprandi, 5-Apr-2020.)
(𝜑𝑆 ∈ {ℝ, ℂ})    &   (𝜑𝑋 ∈ ((TopOpen‘ℂfld) ↾t 𝑆))    &   (𝜑𝑃 ∈ ℕ)    &   (𝜑𝑀 ∈ ℕ0)    &   𝐹 = (𝑥𝑋 ↦ ((𝑥↑(𝑃 − 1)) · ∏𝑗 ∈ (1...𝑀)((𝑥𝑗)↑𝑃)))    &   (𝜑𝑁 ∈ ℕ0)    &   𝐻 = (𝑗 ∈ (0...𝑀) ↦ (𝑥𝑋 ↦ ((𝑥𝑗)↑if(𝑗 = 0, (𝑃 − 1), 𝑃))))    &   (𝜑𝐽𝑋)    &   (𝜑𝐽 ∈ ℤ)    &   𝐶 = (𝑛 ∈ ℕ0 ↦ {𝑐 ∈ ((0...𝑛) ↑𝑚 (0...𝑀)) ∣ Σ𝑗 ∈ (0...𝑀)(𝑐𝑗) = 𝑛})       (𝜑 → (((𝑆 D𝑛 𝐹)‘𝑁)‘𝐽) ∈ ℤ)

Theoremetransclem37 40806* (𝑃 − 1) factorial divides the 𝑁-th derivative of 𝐹 applied to 𝐽. (Contributed by Glauco Siliprandi, 5-Apr-2020.)
(𝜑𝑆 ∈ {ℝ, ℂ})    &   (𝜑𝑋 ∈ ((TopOpen‘ℂfld) ↾t 𝑆))    &   (𝜑𝑃 ∈ ℕ)    &   (𝜑𝑀 ∈ ℕ0)    &   𝐹 = (𝑥𝑋 ↦ ((𝑥↑(𝑃 − 1)) · ∏𝑗 ∈ (1...𝑀)((𝑥𝑗)↑𝑃)))    &   (𝜑𝑁 ∈ ℕ0)    &   𝐻 = (𝑗 ∈ (0...𝑀) ↦ (𝑥𝑋 ↦ ((𝑥𝑗)↑if(𝑗 = 0, (𝑃 − 1), 𝑃))))    &   𝐶 = (𝑛 ∈ ℕ0 ↦ {𝑐 ∈ ((0...𝑛) ↑𝑚 (0...𝑀)) ∣ Σ𝑗 ∈ (0...𝑀)(𝑐𝑗) = 𝑛})    &   (𝜑𝐽 ∈ (0...𝑀))    &   (𝜑𝐽𝑋)       (𝜑 → (!‘(𝑃 − 1)) ∥ (((𝑆 D𝑛 𝐹)‘𝑁)‘𝐽))

Theoremetransclem38 40807* 𝑃 divides the I -th derivative of 𝐹 applied to 𝐽. if it is not the case that 𝐼 = 𝑃 − 1 and 𝐽 = 0. This is case 1 and the second part of case 2 proven in in [Juillerat] p. 13 . (Contributed by Glauco Siliprandi, 5-Apr-2020.)
(𝜑𝑃 ∈ ℕ)    &   (𝜑𝑀 ∈ ℕ0)    &   𝐹 = (𝑥 ∈ ℝ ↦ ((𝑥↑(𝑃 − 1)) · ∏𝑗 ∈ (1...𝑀)((𝑥𝑗)↑𝑃)))    &   (𝜑𝐼 ∈ ℕ0)    &   (𝜑𝐽 ∈ (0...𝑀))    &   (𝜑 → ¬ (𝐼 = (𝑃 − 1) ∧ 𝐽 = 0))    &   𝐶 = (𝑛 ∈ ℕ0 ↦ {𝑐 ∈ ((0...𝑛) ↑𝑚 (0...𝑀)) ∣ Σ𝑗 ∈ (0...𝑀)(𝑐𝑗) = 𝑛})       (𝜑𝑃 ∥ ((((ℝ D𝑛 𝐹)‘𝐼)‘𝐽) / (!‘(𝑃 − 1))))

Theoremetransclem39 40808* 𝐺 is a function. (Contributed by Glauco Siliprandi, 5-Apr-2020.)
(𝜑𝑃 ∈ ℕ)    &   (𝜑𝑀 ∈ ℕ0)    &   𝐹 = (𝑥 ∈ ℝ ↦ ((𝑥↑(𝑃 − 1)) · ∏𝑗 ∈ (1...𝑀)((𝑥𝑗)↑𝑃)))    &   𝐺 = (𝑥 ∈ ℝ ↦ Σ𝑖 ∈ (0...𝑅)(((ℝ D𝑛 𝐹)‘𝑖)‘𝑥))       (𝜑𝐺:ℝ⟶ℂ)

Theoremetransclem40 40809* The 𝑁-th derivative of 𝐹 is continuous. (Contributed by Glauco Siliprandi, 5-Apr-2020.)
(𝜑𝑆 ∈ {ℝ, ℂ})    &   (𝜑𝑋 ∈ ((TopOpen‘ℂfld) ↾t 𝑆))    &   (𝜑𝑃 ∈ ℕ)    &   (𝜑𝑀 ∈ ℕ0)    &   𝐹 = (𝑥𝑋 ↦ ((𝑥↑(𝑃 − 1)) · ∏𝑘 ∈ (1...𝑀)((𝑥𝑘)↑𝑃)))    &   (𝜑𝑁 ∈ ℕ0)       (𝜑 → ((𝑆 D𝑛 𝐹)‘𝑁) ∈ (𝑋cn→ℂ))

Theoremetransclem41 40810* 𝑃 does not divide the P-1 -th derivative of 𝐹 applied to 0. This is the first part of case 2: proven in in [Juillerat] p. 13 . (Contributed by Glauco Siliprandi, 5-Apr-2020.)
(𝜑𝑀 ∈ ℕ0)    &   (𝜑𝑃 ∈ ℙ)    &   (𝜑 → (!‘𝑀) < 𝑃)    &   𝐹 = (𝑥 ∈ ℝ ↦ ((𝑥↑(𝑃 − 1)) · ∏𝑗 ∈ (1...𝑀)((𝑥𝑗)↑𝑃)))       (𝜑 → ¬ 𝑃 ∥ ((((ℝ D𝑛 𝐹)‘(𝑃 − 1))‘0) / (!‘(𝑃 − 1))))

Theoremetransclem42 40811* The 𝑁-th derivative of 𝐹 applied to 𝐽 is an integer. (Contributed by Glauco Siliprandi, 5-Apr-2020.)
(𝜑𝑆 ∈ {ℝ, ℂ})    &   (𝜑𝑋 ∈ ((TopOpen‘ℂfld) ↾t 𝑆))    &   (𝜑𝑃 ∈ ℕ)    &   (𝜑𝑀 ∈ ℕ0)    &   𝐹 = (𝑥𝑋 ↦ ((𝑥↑(𝑃 − 1)) · ∏𝑗 ∈ (1...𝑀)((𝑥𝑗)↑𝑃)))    &   (𝜑𝑁 ∈ ℕ0)    &   (𝜑𝐽𝑋)    &   (𝜑𝐽 ∈ ℤ)       (𝜑 → (((𝑆 D𝑛 𝐹)‘𝑁)‘𝐽) ∈ ℤ)

Theoremetransclem43 40812* 𝐺 is a continuous function. (Contributed by Glauco Siliprandi, 5-Apr-2020.)
(𝜑𝑆 ∈ {ℝ, ℂ})    &   (𝜑𝑋 ∈ ((TopOpen‘ℂfld) ↾t 𝑆))    &   (𝜑𝑃 ∈ ℕ)    &   (𝜑𝑀 ∈ ℕ0)    &   𝐹 = (𝑥𝑋 ↦ ((𝑥↑(𝑃 − 1)) · ∏𝑗 ∈ (1...𝑀)((𝑥𝑗)↑𝑃)))    &   𝐺 = (𝑥𝑋 ↦ Σ𝑖 ∈ (0...𝑅)(((𝑆 D𝑛 𝐹)‘𝑖)‘𝑥))       (𝜑𝐺 ∈ (𝑋cn→ℂ))

Theoremetransclem44 40813* The given finite sum is nonzero. This is the claim proved after equation (7) in [Juillerat] p. 12 . (Contributed by Glauco Siliprandi, 5-Apr-2020.)
(𝜑𝐴:ℕ0⟶ℤ)    &   (𝜑 → (𝐴‘0) ≠ 0)    &   (𝜑𝑀 ∈ ℕ0)    &   (𝜑𝑃 ∈ ℙ)    &   (𝜑 → (abs‘(𝐴‘0)) < 𝑃)    &   (𝜑 → (!‘𝑀) < 𝑃)    &   𝐹 = (𝑥 ∈ ℝ ↦ ((𝑥↑(𝑃 − 1)) · ∏𝑗 ∈ (1...𝑀)((𝑥𝑗)↑𝑃)))    &   𝐾 = (Σ𝑘 ∈ ((0...𝑀) × (0...((𝑀 · 𝑃) + (𝑃 − 1))))((𝐴‘(1st𝑘)) · (((ℝ D𝑛 𝐹)‘(2nd𝑘))‘(1st𝑘))) / (!‘(𝑃 − 1)))       (𝜑𝐾 ≠ 0)

Theoremetransclem45 40814* 𝐾 is an integer. (Contributed by Glauco Siliprandi, 5-Apr-2020.)
(𝜑𝑃 ∈ ℕ)    &   (𝜑𝑀 ∈ ℕ0)    &   𝐹 = (𝑥 ∈ ℝ ↦ ((𝑥↑(𝑃 − 1)) · ∏𝑗 ∈ (1...𝑀)((𝑥𝑗)↑𝑃)))    &   (𝜑𝐴:ℕ0⟶ℤ)    &   𝐾 = (Σ𝑘 ∈ ((0...𝑀) × (0...𝑅))((𝐴‘(1st𝑘)) · (((ℝ D𝑛 𝐹)‘(2nd𝑘))‘(1st𝑘))) / (!‘(𝑃 − 1)))       (𝜑𝐾 ∈ ℤ)

Theoremetransclem46 40815* This is the proof for equation *(7) in [Juillerat] p. 12. The proven equality will lead to a contradiction, because the left-hand side goes to 0 for large 𝑃, but the right-hand side is a nonzero integer. (Contributed by Glauco Siliprandi, 5-Apr-2020.)
(𝜑𝑄 ∈ ((Poly‘ℤ) ∖ {0𝑝}))    &   (𝜑 → (𝑄‘e) = 0)    &   𝐴 = (coeff‘𝑄)    &   𝑀 = (deg‘𝑄)    &   (𝜑 → ℝ ⊆ ℝ)    &   (𝜑 → ℝ ∈ {ℝ, ℂ})    &   (𝜑 → ℝ ∈ ((TopOpen‘ℂfld) ↾t ℝ))    &   (𝜑𝑃 ∈ ℕ)    &   𝐹 = (𝑥 ∈ ℝ ↦ ((𝑥↑(𝑃 − 1)) · ∏𝑗 ∈ (1...𝑀)((𝑥𝑗)↑𝑃)))    &   𝐿 = Σ𝑗 ∈ (0...𝑀)(((𝐴𝑗) · (e↑𝑐𝑗)) · ∫(0(,)𝑗)((e↑𝑐-𝑥) · (𝐹𝑥)) d𝑥)    &   𝑅 = ((𝑀 · 𝑃) + (𝑃 − 1))    &   𝐺 = (𝑥 ∈ ℝ ↦ Σ𝑖 ∈ (0...𝑅)(((ℝ D𝑛 𝐹)‘𝑖)‘𝑥))    &   𝑂 = (𝑥 ∈ (0[,]𝑗) ↦ -((e↑𝑐-𝑥) · (𝐺𝑥)))       (𝜑 → (𝐿 / (!‘(𝑃 − 1))) = (-Σ𝑘 ∈ ((0...𝑀) × (0...𝑅))((𝐴‘(1st𝑘)) · (((ℝ D𝑛 𝐹)‘(2nd𝑘))‘(1st𝑘))) / (!‘(𝑃 − 1))))

Theoremetransclem47 40816* e is transcendental. Section *5 of [Juillerat] p. 11 can be used as a reference for this proof. (Contributed by Glauco Siliprandi, 5-Apr-2020.)
(𝜑𝑄 ∈ ((Poly‘ℤ) ∖ {0𝑝}))    &   (𝜑 → (𝑄‘e) = 0)    &   𝐴 = (coeff‘𝑄)    &   (𝜑 → (𝐴‘0) ≠ 0)    &   𝑀 = (deg‘𝑄)    &   (𝜑𝑃 ∈ ℙ)    &   (𝜑 → (abs‘(𝐴‘0)) < 𝑃)    &   (𝜑 → (!‘𝑀) < 𝑃)    &   (𝜑 → (Σ𝑗 ∈ (0...𝑀)((abs‘((𝐴𝑗) · (e↑𝑐𝑗))) · (𝑀 · (𝑀↑(𝑀 + 1)))) · (((𝑀↑(𝑀 + 1))↑(𝑃 − 1)) / (!‘(𝑃 − 1)))) < 1)    &   𝐹 = (𝑥 ∈ ℝ ↦ ((𝑥↑(𝑃 − 1)) · ∏𝑗 ∈ (1...𝑀)((𝑥𝑗)↑𝑃)))    &   𝐿 = Σ𝑗 ∈ (0...𝑀)(((𝐴𝑗) · (e↑𝑐𝑗)) · ∫(0(,)𝑗)((e↑𝑐-𝑥) · (𝐹𝑥)) d𝑥)    &   𝐾 = (𝐿 / (!‘(𝑃 − 1)))       (𝜑 → ∃𝑘 ∈ ℤ (𝑘 ≠ 0 ∧ (abs‘𝑘) < 1))

Theoremetransclem48 40817* e is transcendental. Section *5 of [Juillerat] p. 11 can be used as a reference for this proof. In this lemma, a large enough prime 𝑝 is chosen: it will be used by subsequent lemmas. (Contributed by Glauco Siliprandi, 5-Apr-2020.) (Revised by AV, 28-Sep-2020.)
(𝜑𝑄 ∈ ((Poly‘ℤ) ∖ {0𝑝}))    &   (𝜑 → (𝑄‘e) = 0)    &   𝐴 = (coeff‘𝑄)    &   (𝜑 → (𝐴‘0) ≠ 0)    &   𝑀 = (deg‘𝑄)    &   𝐶 = Σ𝑗 ∈ (0...𝑀)((abs‘((𝐴𝑗) · (e↑𝑐𝑗))) · (𝑀 · (𝑀↑(𝑀 + 1))))    &   𝑆 = (𝑛 ∈ ℕ0 ↦ (𝐶 · (((𝑀↑(𝑀 + 1))↑𝑛) / (!‘𝑛))))    &   𝐼 = inf({𝑖 ∈ ℕ0 ∣ ∀𝑛 ∈ (ℤ𝑖)(abs‘(𝑆𝑛)) < 1}, ℝ, < )    &   𝑇 = sup({(abs‘(𝐴‘0)), (!‘𝑀), 𝐼}, ℝ*, < )       (𝜑 → ∃𝑘 ∈ ℤ (𝑘 ≠ 0 ∧ (abs‘𝑘) < 1))

Theoremetransc 40818 e is transcendental. Section *5 of [Juillerat] p. 11 can be used as a reference for this proof. (Contributed by Glauco Siliprandi, 5-Apr-2020.) (Proof shortened by AV, 28-Sep-2020.)
e ∈ (ℂ ∖ 𝔸)

20.32.18  n-dimensional Euclidean space

Theoremrrxtopn 40819* The topology of the generalized real Euclidean space. (Contributed by Glauco Siliprandi, 24-Dec-2020.)
(𝜑𝐼𝑉)       (𝜑 → (TopOpen‘(ℝ^‘𝐼)) = (MetOpen‘(𝑓 ∈ (Base‘(ℝ^‘𝐼)), 𝑔 ∈ (Base‘(ℝ^‘𝐼)) ↦ (√‘(ℝfld Σg (𝑥𝐼 ↦ (((𝑓𝑥) − (𝑔𝑥))↑2)))))))

Theoremrrxngp 40820 Generalized Euclidean real spaces are normed groups. (Contributed by Glauco Siliprandi, 24-Dec-2020.)
(𝐼𝑉 → (ℝ^‘𝐼) ∈ NrmGrp)

Theoremrrxbasefi 40821 The base of the generalized real Euclidean space, when the dimension of the space is finite. This justifies the use of (ℝ ↑𝑚 𝑋) for the development of the Lebeasgue measure theory for n-dimensional Real numbers. (Contributed by Glauco Siliprandi, 24-Dec-2020.)
(𝜑𝑋 ∈ Fin)    &   𝐻 = (ℝ^‘𝑋)    &   𝐵 = (Base‘𝐻)       (𝜑𝐵 = (ℝ ↑𝑚 𝑋))

Theoremrrxtps 40822 Generalized Euclidean real spaces are topological spaces. (Contributed by Glauco Siliprandi, 24-Dec-2020.)
(𝐼𝑉 → (ℝ^‘𝐼) ∈ TopSp)

Theoremrrxdsfi 40823* The distance over generalized Euclidean spaces. Finite dimensional case. (Contributed by Glauco Siliprandi, 24-Dec-2020.)
𝐻 = (ℝ^‘𝐼)    &   𝐵 = (ℝ ↑𝑚 𝐼)       (𝐼 ∈ Fin → (dist‘𝐻) = (𝑓𝐵, 𝑔𝐵 ↦ (√‘Σ𝑘𝐼 (((𝑓𝑘) − (𝑔𝑘))↑2))))

Theoremrrxtopnfi 40824* The topology of the n-dimensional real Euclidean space. (Contributed by Glauco Siliprandi, 24-Dec-2020.)
(𝜑𝐼 ∈ Fin)       (𝜑 → (TopOpen‘(ℝ^‘𝐼)) = (MetOpen‘(𝑓 ∈ (ℝ ↑𝑚 𝐼), 𝑔 ∈ (ℝ ↑𝑚 𝐼) ↦ (√‘Σ𝑘𝐼 (((𝑓𝑘) − (𝑔𝑘))↑2)))))

Theoremrrxmetfi 40825 Euclidean space is a metric space. Finite dimensional version. (Contributed by Glauco Siliprandi, 24-Dec-2020.)
𝐷 = (dist‘(ℝ^‘𝐼))       (𝐼 ∈ Fin → 𝐷 ∈ (Met‘(ℝ ↑𝑚 𝐼)))

Theoremrrxtopon 40826 The topology on Generalized Euclidean real spaces. (Contributed by Glauco Siliprandi, 24-Dec-2020.)
𝐽 = (TopOpen‘(ℝ^‘𝐼))       (𝐼𝑉𝐽 ∈ (TopOn‘(Base‘(ℝ^‘𝐼))))

Theoremrrxtop 40827 The topology on Generalized Euclidean real spaces. (Contributed by Glauco Siliprandi, 24-Dec-2020.)
𝐽 = (TopOpen‘(ℝ^‘𝐼))       (𝐼𝑉𝐽 ∈ Top)

Theoremrrndistlt 40828* Given two points in the space of n-dimensional real numbers, if every component is closer than 𝐸 then the distance between the two points is less then ((√‘𝑛) · 𝐸) (Contributed by Glauco Siliprandi, 24-Dec-2020.)
(𝜑𝐼 ∈ Fin)    &   (𝜑𝐼 ≠ ∅)    &   𝑁 = (#‘𝐼)    &   (𝜑𝑋 ∈ (ℝ ↑𝑚 𝐼))    &   (𝜑𝑌 ∈ (ℝ ↑𝑚 𝐼))    &   ((𝜑𝑖𝐼) → (abs‘((𝑋𝑖) − (𝑌𝑖))) < 𝐸)    &   (𝜑𝐸 ∈ ℝ+)    &   𝐷 = (dist‘(ℝ^‘𝐼))       (𝜑 → (𝑋𝐷𝑌) < ((√‘𝑁) · 𝐸))

Theoremrrxtoponfi 40829 The topology on n-dimensional Euclidean real spaces. (Contributed by Glauco Siliprandi, 24-Dec-2020.)
𝐽 = (TopOpen‘(ℝ^‘𝐼))       (𝐼 ∈ Fin → 𝐽 ∈ (TopOn‘(ℝ ↑𝑚 𝐼)))

Theoremrrxunitopnfi 40830 The base set of the standard topology on the space of n-dimensional Real numbers. (Contributed by Glauco Siliprandi, 24-Dec-2020.)
(𝑋 ∈ Fin → (TopOpen‘(ℝ^‘𝑋)) = (ℝ ↑𝑚 𝑋))

Theoremrrxtopn0 40831 The topology of the zero-dimensional real Euclidean space. (Contributed by Glauco Siliprandi, 24-Dec-2020.)
(TopOpen‘(ℝ^‘∅)) = 𝒫 {∅}

Theoremqndenserrnbllem 40832* n-dimensional rational numbers are dense in the space of n-dimensional real numbers, with respect to the n-dimensional standard topology. (Contributed by Glauco Siliprandi, 24-Dec-2020.)
(𝜑𝐼 ∈ Fin)    &   (𝜑𝐼 ≠ ∅)    &   (𝜑𝑋 ∈ (ℝ ↑𝑚 𝐼))    &   𝐷 = (dist‘(ℝ^‘𝐼))    &   (𝜑𝐸 ∈ ℝ+)       (𝜑 → ∃𝑦 ∈ (ℚ ↑𝑚 𝐼)𝑦 ∈ (𝑋(ball‘𝐷)𝐸))

Theoremqndenserrnbl 40833* n-dimensional rational numbers are dense in the space of n-dimensional real numbers, with respect to the n-dimensional standard topology. (Contributed by Glauco Siliprandi, 24-Dec-2020.)
(𝜑𝐼 ∈ Fin)    &   (𝜑𝑋 ∈ (ℝ ↑𝑚 𝐼))    &   𝐷 = (dist‘(ℝ^‘𝐼))    &   (𝜑𝐸 ∈ ℝ+)       (𝜑 → ∃𝑦 ∈ (ℚ ↑𝑚 𝐼)𝑦 ∈ (𝑋(ball‘𝐷)𝐸))

Theoremrrxtopn0b 40834 The topology of the zero-dimensional real Euclidean space. (Contributed by Glauco Siliprandi, 24-Dec-2020.)
(TopOpen‘(ℝ^‘∅)) = {∅, {∅}}

Theoremqndenserrnopnlem 40835* n-dimensional rational numbers are dense in the space of n-dimensional real numbers, with respect to the n-dimensional standard topology. (Contributed by Glauco Siliprandi, 24-Dec-2020.)
(𝜑𝐼 ∈ Fin)    &   𝐽 = (TopOpen‘(ℝ^‘𝐼))    &   (𝜑𝑉𝐽)    &   (𝜑𝑋𝑉)    &   𝐷 = (dist‘(ℝ^‘𝐼))       (𝜑 → ∃𝑦 ∈ (ℚ ↑𝑚 𝐼)𝑦𝑉)

Theoremqndenserrnopn 40836* n-dimensional rational numbers are dense in the space of n-dimensional real numbers, with respect to the n-dimensional standard topology. (Contributed by Glauco Siliprandi, 24-Dec-2020.)
(𝜑𝐼 ∈ Fin)    &   𝐽 = (TopOpen‘(ℝ^‘𝐼))    &   (𝜑𝑉𝐽)    &   (𝜑𝑉 ≠ ∅)       (𝜑 → ∃𝑦 ∈ (ℚ ↑𝑚 𝐼)𝑦𝑉)

Theoremqndenserrn 40837 n-dimensional rational numbers are dense in the space of n-dimensional real numbers, with respect to the n-dimensional standard topology. (Contributed by Glauco Siliprandi, 24-Dec-2020.)
(𝜑𝐼 ∈ Fin)    &   𝐽 = (TopOpen‘(ℝ^‘𝐼))       (𝜑 → ((cls‘𝐽)‘(ℚ ↑𝑚 𝐼)) = (ℝ ↑𝑚 𝐼))

Theoremrrxsnicc 40838* A multidimensional singleton expressed as a multidimensional closed interval. (Contributed by Glauco Siliprandi, 8-Apr-2021.)
(𝜑𝐴 ∈ (ℝ ↑𝑚 𝑋))       (𝜑X𝑘𝑋 ((𝐴𝑘)[,](𝐴𝑘)) = {𝐴})

Theoremrrnprjdstle 40839 The distance between two points in Euclidean space is greater than the distance between the projections onto one coordinate. (Contributed by Glauco Siliprandi, 8-Apr-2021.)
(𝜑𝑋 ∈ Fin)    &   (𝜑𝐹:𝑋⟶ℝ)    &   (𝜑𝐺:𝑋⟶ℝ)    &   (𝜑𝐼𝑋)    &   𝐷 = (dist‘(ℝ^‘𝑋))       (𝜑 → (abs‘((𝐹𝐼) − (𝐺𝐼))) ≤ (𝐹𝐷𝐺))

Theoremrrndsmet 40840* 𝐷 is a metric for the n-dimensional real Euclidean space. (Contributed by Glauco Siliprandi, 8-Apr-2021.)
(𝜑𝑋 ∈ Fin)    &   𝐷 = (𝑓 ∈ (ℝ ↑𝑚 𝑋), 𝑔 ∈ (ℝ ↑𝑚 𝑋) ↦ (√‘Σ𝑘𝑋 (((𝑓𝑘) − (𝑔𝑘))↑2)))       (𝜑𝐷 ∈ (Met‘(ℝ ↑𝑚 𝑋)))

Theoremrrndsxmet 40841* 𝐷 is an extended metric for the n-dimensional real Euclidean space. (Contributed by Glauco Siliprandi, 8-Apr-2021.)
(𝜑𝑋 ∈ Fin)    &   𝐷 = (𝑓 ∈ (ℝ ↑𝑚 𝑋), 𝑔 ∈ (ℝ ↑𝑚 𝑋) ↦ (√‘Σ𝑘𝑋 (((𝑓𝑘) − (𝑔𝑘))↑2)))       (𝜑𝐷 ∈ (∞Met‘(ℝ ↑𝑚 𝑋)))

Theoremioorrnopnlem 40842* The a point in an indexed product of open intervals is contained in an open ball that is contained in the indexed product of open intervals. (Contributed by Glauco Siliprandi, 8-Apr-2021.)
(𝜑𝑋 ∈ Fin)    &   (𝜑𝑋 ≠ ∅)    &   (𝜑𝐴:𝑋⟶ℝ)    &   (𝜑𝐵:𝑋⟶ℝ)    &   (𝜑𝐹X𝑖𝑋 ((𝐴𝑖)(,)(𝐵𝑖)))    &   𝐻 = ran (𝑖𝑋 ↦ if(((𝐵𝑖) − (𝐹𝑖)) ≤ ((𝐹𝑖) − (𝐴𝑖)), ((𝐵𝑖) − (𝐹𝑖)), ((𝐹𝑖) − (𝐴𝑖))))    &   𝐸 = inf(𝐻, ℝ, < )    &   𝑉 = (𝐹(ball‘𝐷)𝐸)    &   𝐷 = (𝑓 ∈ (ℝ ↑𝑚 𝑋), 𝑔 ∈ (ℝ ↑𝑚 𝑋) ↦ (√‘Σ𝑘𝑋 (((𝑓𝑘) − (𝑔𝑘))↑2)))       (𝜑 → ∃𝑣 ∈ (TopOpen‘(ℝ^‘𝑋))(𝐹𝑣𝑣X𝑖𝑋 ((𝐴𝑖)(,)(𝐵𝑖))))

Theoremioorrnopn 40843* The indexed product of open intervals is an open set in (ℝ^‘𝑋). (Contributed by Glauco Siliprandi, 8-Apr-2021.)
(𝜑𝑋 ∈ Fin)    &   (𝜑𝐴:𝑋⟶ℝ)    &   (𝜑𝐵:𝑋⟶ℝ)       (𝜑X𝑖𝑋 ((𝐴𝑖)(,)(𝐵𝑖)) ∈ (TopOpen‘(ℝ^‘𝑋)))

Theoremioorrnopnxrlem 40844* Given a point 𝐹 that belongs to an indexed product of (possibly unbounded) open intervals, then 𝐹 belongs to an open product of bounded open intervals that's a subset of the original indexed product. (Contributed by Glauco Siliprandi, 8-Apr-2021.)
(𝜑𝑋 ∈ Fin)    &   (𝜑𝐴:𝑋⟶ℝ*)    &   (𝜑𝐵:𝑋⟶ℝ*)    &   (𝜑𝐹X𝑖𝑋 ((𝐴𝑖)(,)(𝐵𝑖)))    &   𝐿 = (𝑖𝑋 ↦ if((𝐴𝑖) = -∞, ((𝐹𝑖) − 1), (𝐴𝑖)))    &   𝑅 = (𝑖𝑋 ↦ if((𝐵𝑖) = +∞, ((𝐹𝑖) + 1), (𝐵𝑖)))    &   𝑉 = X𝑖𝑋 ((𝐿𝑖)(,)(𝑅𝑖))       (𝜑 → ∃𝑣 ∈ (TopOpen‘(ℝ^‘𝑋))(𝐹𝑣𝑣X𝑖𝑋 ((𝐴𝑖)(,)(𝐵𝑖))))

Theoremioorrnopnxr 40845* The indexed product of open intervals is an open set in (ℝ^‘𝑋). Similar to ioorrnopn 40843 but here unbounded intervals are allowed. (Contributed by Glauco Siliprandi, 8-Apr-2021.)
(𝜑𝑋 ∈ Fin)    &   (𝜑𝐴:𝑋⟶ℝ*)    &   (𝜑𝐵:𝑋⟶ℝ*)       (𝜑X𝑖𝑋 ((𝐴𝑖)(,)(𝐵𝑖)) ∈ (TopOpen‘(ℝ^‘𝑋)))

20.32.19  Basic measure theory

20.32.19.1  σ-Algebras

Proofs for most of the theorems in section 111 of [Fremlin1]

Syntaxcsalg 40846 Extend class notation with the class of all sigma-algebras.
class SAlg

Definitiondf-salg 40847* Define the class of sigma-algebras. (Contributed by Glauco Siliprandi, 17-Aug-2020.)
SAlg = {𝑥 ∣ (∅ ∈ 𝑥 ∧ ∀𝑦𝑥 ( 𝑥𝑦) ∈ 𝑥 ∧ ∀𝑦 ∈ 𝒫 𝑥(𝑦 ≼ ω → 𝑦𝑥))}

Syntaxcsalon 40848 Extend class notation with the class of sigma-algebras on a set.
class SalOn

Definitiondf-salon 40849* Define the set of sigma-algebra on a given set. (Contributed by Glauco Siliprandi, 17-Aug-2020.)
SalOn = (𝑥 ∈ V ↦ {𝑠 ∈ SAlg ∣ 𝑠 = 𝑥})

Syntaxcsalgen 40850 Extend class notation with the class of sigma-algebra generator.
class SalGen

Definitiondf-salgen 40851* Define the sigma-algebra generated by a given set. Definition 111G (b) of [Fremlin1] p. 13. The sigma-algebra generated by a set is the smallest sigma-algebra, on the same base set, that includes the set, see dfsalgen2 40877. The base set of the sigma-algebras used for the intersection needs to be the same, otherwise the resulting set is not guaranteed to be a sigma-algebra, as shown in the counterexample salgencntex 40879. (Contributed by Glauco Siliprandi, 17-Aug-2020.) (Revised by Glauco Siliprandi, 1-Jan-2021.)
SalGen = (𝑥 ∈ V ↦ {𝑠 ∈ SAlg ∣ ( 𝑠 = 𝑥𝑥𝑠)})

Theoremissal 40852* Express the predicate "𝑆 is a sigma-algebra." (Contributed by Glauco Siliprandi, 17-Aug-2020.)
(𝑆𝑉 → (𝑆 ∈ SAlg ↔ (∅ ∈ 𝑆 ∧ ∀𝑦𝑆 ( 𝑆𝑦) ∈ 𝑆 ∧ ∀𝑦 ∈ 𝒫 𝑆(𝑦 ≼ ω → 𝑦𝑆))))

Theorempwsal 40853 The power set of a given set is a sigma-algebra (the so called discrete sigma-algebra). (Contributed by Glauco Siliprandi, 17-Aug-2020.)
(𝑋𝑉 → 𝒫 𝑋 ∈ SAlg)

Theoremsalunicl 40854 SAlg sigma-algebra is closed under countable union. (Contributed by Glauco Siliprandi, 17-Aug-2020.)
(𝜑𝑆 ∈ SAlg)    &   (𝜑𝑇 ∈ 𝒫 𝑆)    &   (𝜑𝑇 ≼ ω)       (𝜑 𝑇𝑆)

Theoremsaluncl 40855 The union of two sets in a sigma-algebra is in the sigma-algebra. (Contributed by Glauco Siliprandi, 17-Aug-2020.)
((𝑆 ∈ SAlg ∧ 𝐸𝑆𝐹𝑆) → (𝐸𝐹) ∈ 𝑆)

Theoremprsal 40856 The pair of the empty set and the whole base is a sigma-algebra. (Contributed by Glauco Siliprandi, 17-Aug-2020.)
(𝑋𝑉 → {∅, 𝑋} ∈ SAlg)

Theoremsaldifcl 40857 The complement of an element of a sigma-algebra is in the sigma-algebra. (Contributed by Glauco Siliprandi, 17-Aug-2020.)
((𝑆 ∈ SAlg ∧ 𝐸𝑆) → ( 𝑆𝐸) ∈ 𝑆)

Theorem0sal 40858 The empty set belongs to every sigma-algebra. (Contributed by Glauco Siliprandi, 17-Aug-2020.)
(𝑆 ∈ SAlg → ∅ ∈ 𝑆)

Theoremsalgenval 40859* The sigma-algebra generated by a set. (Contributed by Glauco Siliprandi, 3-Jan-2021.)
(𝑋𝑉 → (SalGen‘𝑋) = {𝑠 ∈ SAlg ∣ ( 𝑠 = 𝑋𝑋𝑠)})

Theoremsaliuncl 40860* SAlg sigma-algebra is closed under countable indexed union. (Contributed by Glauco Siliprandi, 17-Aug-2020.)
(𝜑𝑆 ∈ SAlg)    &   (𝜑𝐾 ≼ ω)    &   ((𝜑𝑘𝐾) → 𝐸𝑆)       (𝜑 𝑘𝐾 𝐸𝑆)

Theoremsalincl 40861 The intersection of two sets in a sigma-algebra is in the sigma-algebra. (Contributed by Glauco Siliprandi, 17-Aug-2020.)
((𝑆 ∈ SAlg ∧ 𝐸𝑆𝐹𝑆) → (𝐸𝐹) ∈ 𝑆)

Theoremsaluni 40862 A set is an element of any sigma-algebra on it . (Contributed by Glauco Siliprandi, 17-Aug-2020.)
(𝑆 ∈ SAlg → 𝑆𝑆)

Theoremsaliincl 40863* SAlg sigma-algebra is closed under countable indexed intersection. (Contributed by Glauco Siliprandi, 17-Aug-2020.)
(𝜑𝑆 ∈ SAlg)    &   (𝜑𝐾 ≼ ω)    &   (𝜑𝐾 ≠ ∅)    &   ((𝜑𝑘𝐾) → 𝐸𝑆)       (𝜑 𝑘𝐾 𝐸𝑆)

Theoremsaldifcl2 40864 The difference of two elements of a sigma-algebra is in the sigma-algebra. (Contributed by Glauco Siliprandi, 17-Aug-2020.)
((𝑆 ∈ SAlg ∧ 𝐸𝑆𝐹𝑆) → (𝐸𝐹) ∈ 𝑆)

Theoremintsaluni 40865* The union of an arbitrary intersection of sigma-algebras on the same set 𝑋, is 𝑋. (Contributed by Glauco Siliprandi, 17-Aug-2020.)
(𝜑𝐺 ⊆ SAlg)    &   (𝜑𝐺 ≠ ∅)    &   ((𝜑𝑠𝐺) → 𝑠 = 𝑋)       (𝜑 𝐺 = 𝑋)

Theoremintsal 40866* The arbitrary intersection of sigma-algebra (on the same set 𝑋) is a sigma-algebra ( on the same set 𝑋, see intsaluni 40865). (Contributed by Glauco Siliprandi, 17-Aug-2020.)
(𝜑𝐺 ⊆ SAlg)    &   (𝜑𝐺 ≠ ∅)    &   ((𝜑𝑠𝐺) → 𝑠 = 𝑋)       (𝜑 𝐺 ∈ SAlg)

Theoremsalgenn0 40867* The set used in the definition of the generated sigma-algebra, is not empty. (Contributed by Glauco Siliprandi, 3-Jan-2021.)
(𝑋𝑉 → {𝑠 ∈ SAlg ∣ ( 𝑠 = 𝑋𝑋𝑠)} ≠ ∅)

Theoremsalgencl 40868 SalGen actually generates a sigma-algebra. (Contributed by Glauco Siliprandi, 3-Jan-2021.)
(𝑋𝑉 → (SalGen‘𝑋) ∈ SAlg)

Theoremissald 40869* Sufficient condition to prove that 𝑆 is sigma-algebra. (Contributed by Glauco Siliprandi, 3-Jan-2021.)
(𝜑𝑆𝑉)    &   (𝜑 → ∅ ∈ 𝑆)    &   𝑋 = 𝑆    &   ((𝜑𝑦𝑆) → (𝑋𝑦) ∈ 𝑆)    &   ((𝜑𝑦 ∈ 𝒫 𝑆𝑦 ≼ ω) → 𝑦𝑆)       (𝜑𝑆 ∈ SAlg)

Theoremsalexct 40870* An example of non trivial sigma-algebra: the collection of all subsets which either are countable or have countable complement. (Contributed by Glauco Siliprandi, 3-Jan-2021.)
(𝜑𝐴𝑉)    &   𝑆 = {𝑥 ∈ 𝒫 𝐴 ∣ (𝑥 ≼ ω ∨ (𝐴𝑥) ≼ ω)}       (𝜑𝑆 ∈ SAlg)

Theoremsssalgen 40871 A set is a subset of the sigma-algebra it generates. (Contributed by Glauco Siliprandi, 3-Jan-2021.)
𝑆 = (SalGen‘𝑋)       (𝑋𝑉𝑋𝑆)

Theoremsalgenss 40872 The sigma-algebra generated by a set is the smallest sigma-algebra, on the same base set, that includes the set. Proposition 111G (b) of [Fremlin1] p. 13. Notice that the condition "on the same base set" is needed, see the counterexample salgensscntex 40880, where a sigma-algebra is shown that includes a set, but does not include the sigma-algebra generated (the key is that its base set is larger than the base set of the generating set). (Contributed by Glauco Siliprandi, 3-Jan-2021.)
(𝜑𝑋𝑉)    &   𝐺 = (SalGen‘𝑋)    &   (𝜑𝑆 ∈ SAlg)    &   (𝜑𝑋𝑆)    &   (𝜑 𝑆 = 𝑋)       (𝜑𝐺𝑆)

Theoremsalgenuni 40873 The base set of the sigma-algebra generated by a set is the union of the set itself. (Contributed by Glauco Siliprandi, 3-Jan-2021.)
(𝜑𝑋𝑉)    &   𝑆 = (SalGen‘𝑋)    &   𝑈 = 𝑋       (𝜑 𝑆 = 𝑈)

Theoremissalgend 40874* One side of dfsalgen2 40877. If a sigma-algebra on 𝑋 includes 𝑋 and it is included in all the sigma-algebras with such two properties, then it is the sigma-algebra generated by 𝑋. (Contributed by Glauco Siliprandi, 3-Jan-2021.)
(𝜑𝑋𝑉)    &   (𝜑𝑆 ∈ SAlg)    &   (𝜑 𝑆 = 𝑋)    &   (𝜑𝑋𝑆)    &   ((𝜑 ∧ (𝑦 ∈ SAlg ∧ 𝑦 = 𝑋𝑋𝑦)) → 𝑆𝑦)       (𝜑 → (SalGen‘𝑋) = 𝑆)

Theoremsalexct2 40875* An example of a subset that does not belong to a non trivial sigma-algebra, see salexct 40870. (Contributed by Glauco Siliprandi, 3-Jan-2021.)
𝐴 = (0[,]2)    &   𝑆 = {𝑥 ∈ 𝒫 𝐴 ∣ (𝑥 ≼ ω ∨ (𝐴𝑥) ≼ ω)}    &   𝐵 = (0[,]1)        ¬ 𝐵𝑆

Theoremunisalgen 40876 The union of a set belongs to the sigma-algebra generated by the set. (Contributed by Glauco Siliprandi, 3-Jan-2021.)
(𝜑𝑋𝑉)    &   𝑆 = (SalGen‘𝑋)    &   𝑈 = 𝑋       (𝜑𝑈𝑆)

Theoremdfsalgen2 40877* Alternate characterization of the sigma-algebra generated by a set. It is the smallest sigma-algebra, on the same base set, that includes the set. (Contributed by Glauco Siliprandi, 3-Jan-2021.)
(𝜑𝑋𝑉)       (𝜑 → ((SalGen‘𝑋) = 𝑆 ↔ ((𝑆 ∈ SAlg ∧ 𝑆 = 𝑋𝑋𝑆) ∧ ∀𝑦 ∈ SAlg (( 𝑦 = 𝑋𝑋𝑦) → 𝑆𝑦))))

Theoremsalexct3 40878* An example of a sigma-algebra that's not closed under uncountable union. (Contributed by Glauco Siliprandi, 3-Jan-2021.)
𝐴 = (0[,]2)    &   𝑆 = {𝑥 ∈ 𝒫 𝐴 ∣ (𝑥 ≼ ω ∨ (𝐴𝑥) ≼ ω)}    &   𝑋 = ran (𝑦 ∈ (0[,]1) ↦ {𝑦})       (𝑆 ∈ SAlg ∧ 𝑋𝑆 ∧ ¬ 𝑋𝑆)

Theoremsalgencntex 40879* This counterexample shows that df-salgen 40851 needs to require that all containing sigma-algebra have the same base set. Otherwise, the intersection could lead to a set that is not a sigma-algebra. (Contributed by Glauco Siliprandi, 3-Jan-2021.)
𝐴 = (0[,]2)    &   𝑆 = {𝑥 ∈ 𝒫 𝐴 ∣ (𝑥 ≼ ω ∨ (𝐴𝑥) ≼ ω)}    &   𝐵 = (0[,]1)    &   𝑇 = 𝒫 𝐵    &   𝐶 = (𝑆𝑇)    &   𝑍 = {𝑠 ∈ SAlg ∣ 𝐶𝑠}        ¬ 𝑍 ∈ SAlg

Theoremsalgensscntex 40880* This counterexample shows that the sigma-algebra generated by a set is not the smallest sigma-algebra containing the set, if we consider also sigma-algebras with a larger base set. (Contributed by Glauco Siliprandi, 3-Jan-2021.)
𝐴 = (0[,]2)    &   𝑆 = {𝑥 ∈ 𝒫 𝐴 ∣ (𝑥 ≼ ω ∨ (𝐴𝑥) ≼ ω)}    &   𝑋 = ran (𝑦 ∈ (0[,]1) ↦ {𝑦})    &   𝐺 = (SalGen‘𝑋)       (𝑋𝑆𝑆 ∈ SAlg ∧ ¬ 𝐺𝑆)

Theoremissalnnd 40881* Sufficient condition to prove that 𝑆 is sigma-algebra. (Contributed by Glauco Siliprandi, 3-Mar-2021.)
(𝜑𝑆𝑉)    &   (𝜑 → ∅ ∈ 𝑆)    &   𝑋 = 𝑆    &   ((𝜑𝑦𝑆) → (𝑋𝑦) ∈ 𝑆)    &   ((𝜑𝑒:ℕ⟶𝑆) → 𝑛 ∈ ℕ (𝑒𝑛) ∈ 𝑆)       (𝜑𝑆 ∈ SAlg)

Theoremdmvolsal 40882 Lebesgue measurable sets form a sigma-algebra. (Contributed by Glauco Siliprandi, 3-Mar-2021.)
dom vol ∈ SAlg

Theoremsaldifcld 40883 The complement of an element of a sigma-algebra is in the sigma-algebra. (Contributed by Glauco Siliprandi, 26-Jun-2021.)
(𝜑𝑆 ∈ SAlg)    &   (𝜑𝐸𝑆)       (𝜑 → ( 𝑆𝐸) ∈ 𝑆)

Theoremsaluncld 40884 The union of two sets in a sigma-algebra is in the sigma-algebra. (Contributed by Glauco Siliprandi, 26-Jun-2021.)
(𝜑𝑆 ∈ SAlg)    &   (𝜑𝐸𝑆)    &   (𝜑𝐹𝑆)       (𝜑 → (𝐸𝐹) ∈ 𝑆)

Theoremsalgencld 40885 SalGen actually generates a sigma-algebra. (Contributed by Glauco Siliprandi, 26-Jun-2021.)
(𝜑𝑋𝑉)    &   𝑆 = (SalGen‘𝑋)       (𝜑𝑆 ∈ SAlg)

Theorem0sald 40886 The empty set belongs to every sigma-algebra. (Contributed by Glauco Siliprandi, 26-Jun-2021.)
(𝜑𝑆 ∈ SAlg)       (𝜑 → ∅ ∈ 𝑆)

Theoremiooborel 40887 An open interval is a Borel set. (Contributed by Glauco Siliprandi, 26-Jun-2021.)
𝐽 = (topGen‘ran (,))    &   𝐵 = (SalGen‘𝐽)       (𝐴(,)𝐶) ∈ 𝐵

Theoremsalincld 40888 The intersection of two sets in a sigma-algebra is in the sigma-algebra. (Contributed by Glauco Siliprandi, 26-Jun-2021.)
(𝜑𝑆 ∈ SAlg)    &   (𝜑𝐸𝑆)    &   (𝜑𝐹𝑆)       (𝜑 → (𝐸𝐹) ∈ 𝑆)

Theoremsalunid 40889 A set is an element of any sigma-algebra on it . (Contributed by Glauco Siliprandi, 26-Jun-2021.)
(𝜑𝑆 ∈ SAlg)       (𝜑 𝑆𝑆)

Theoremunisalgen2 40890 The union of a set belongs is equal to the union of the sigma-algebra generated by the set. (Contributed by Glauco Siliprandi, 26-Jun-2021.)
(𝜑𝐴𝑉)    &   𝑆 = (SalGen‘𝐴)       (𝜑 𝑆 = 𝐴)

Theorembor1sal 40891 The Borel sigma-algebra on the Reals. (Contributed by Glauco Siliprandi, 26-Jun-2021.)
𝐽 = (topGen‘ran (,))    &   𝐵 = (SalGen‘𝐽)       𝐵 ∈ SAlg

Theoremiocborel 40892 A left-open, right-closed interval is a Borel set. (Contributed by Glauco Siliprandi, 26-Jun-2021.)
(𝜑𝐴 ∈ ℝ*)    &   (𝜑𝐶 ∈ ℝ)    &   𝐽 = (topGen‘ran (,))    &   𝐵 = (SalGen‘𝐽)       (𝜑 → (𝐴(,]𝐶) ∈ 𝐵)

Theoremsubsaliuncllem 40893* A subspace sigma-algebra is closed under countable union. This is Lemma 121A (iii) of [Fremlin1] p. 35. (Contributed by Glauco Siliprandi, 26-Jun-2021.)
𝑦𝜑    &   (𝜑𝑆𝑉)    &   𝐺 = (𝑛 ∈ ℕ ↦ {𝑥𝑆 ∣ (𝐹𝑛) = (𝑥𝐷)})    &   𝐸 = (𝐻𝐺)    &   (𝜑𝐻 Fn ran 𝐺)    &   (𝜑 → ∀𝑦 ∈ ran 𝐺(𝐻𝑦) ∈ 𝑦)       (𝜑 → ∃𝑒 ∈ (𝑆𝑚 ℕ)∀𝑛 ∈ ℕ (𝐹𝑛) = ((𝑒𝑛) ∩ 𝐷))

Theoremsubsaliuncl 40894* A subspace sigma-algebra is closed under countable union. This is Lemma 121A (iii) of [Fremlin1] p. 35. (Contributed by Glauco Siliprandi, 26-Jun-2021.)
(𝜑𝑆 ∈ SAlg)    &   (𝜑𝐷𝑉)    &   𝑇 = (𝑆t 𝐷)    &   (𝜑𝐹:ℕ⟶𝑇)       (𝜑 𝑛 ∈ ℕ (𝐹𝑛) ∈ 𝑇)

Theoremsubsalsal 40895 A subspace sigma-algebra is a sigma algebra. This is Lemma 121A of [Fremlin1] p. 35. (Contributed by Glauco Siliprandi, 26-Jun-2021.)
(𝜑𝑆 ∈ SAlg)    &   (𝜑𝐷𝑉)    &   𝑇 = (𝑆t 𝐷)       (𝜑𝑇 ∈ SAlg)

Theoremsubsaluni 40896 A set belongs to the subspace sigma-algebra it induces. (Contributed by Glauco Siliprandi, 26-Jun-2021.)
(𝜑𝑆 ∈ SAlg)    &   (𝜑𝐴 𝑆)       (𝜑𝐴 ∈ (𝑆t 𝐴))

20.32.19.2  Sum of nonnegative extended reals

Syntaxcsumge0 40897 Extend class notation to include the sum of nonnegative extended reals.
class Σ^

Definitiondf-sumge0 40898* Define the arbitrary sum of nonnegative extended reals. (Contributed by Glauco Siliprandi, 17-Aug-2020.) \$.
Σ^ = (𝑥 ∈ V ↦ if(+∞ ∈ ran 𝑥, +∞, sup(ran (𝑦 ∈ (𝒫 dom 𝑥 ∩ Fin) ↦ Σ𝑤𝑦 (𝑥𝑤)), ℝ*, < )))

Theoremsge0rnre 40899* When Σ^ is applied to nonnegative real numbers the range used in its definition is a subset of the reals. (Contributed by Glauco Siliprandi, 17-Aug-2020.)
(𝜑𝐹:𝑋⟶(0[,)+∞))       (𝜑 → ran (𝑥 ∈ (𝒫 𝑋 ∩ Fin) ↦ Σ𝑦𝑥 (𝐹𝑦)) ⊆ ℝ)

Theoremfge0icoicc 40900 If 𝐹 maps to nonnegative reals, then 𝐹 maps to nonnegative extended reals. (Contributed by Glauco Siliprandi, 17-Aug-2020.)
(𝜑𝐹:𝑋⟶(0[,)+∞))       (𝜑𝐹:𝑋⟶(0[,]+∞))

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